Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
File:H3 633 FC boundary.png Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3[3]} t{3[3,3]}
Coxeter diagrams	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel splitcross.pngFile:CDel branch 11.png ↔ File:CDel node 1.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.pngFile:CDel 3g.pngFile:CDel node g.png ↔ File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png ↔ File:CDel node h0.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node h0.png ↔ File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node h0.png
Cells	{6,3} File:Uniform tiling 63-t0.svg
Faces	hexagon {6}
Edge figure	triangle {3}
Vertex figure	File:Order-3 hexagonal tiling honeycomb verf.png tetrahedron {3,3}
Dual	Order-6 tetrahedral honeycomb
Coxeter groups	${\bar{V}}_3$, [3,3,6] ${\bar{Y}}_3$, [3,6,3] ${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]] ${\overline{PP}}_3$, [3[3,3]]
Properties	Regular

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity. The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.

Images

File:H3 363-1100.png Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H², with horocycles circumscribing vertices of apeirogonal faces.

{6,3,3}	{∞,3}
File:633 honeycomb one cell horosphere.png	File:Order-3 apeirogonal tiling one cell horocycle.png
One hexagonal tiling cell of the hexagonal tiling honeycomb	An order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

File:Hyperbolic subgroup tree 336-direct.png

Subgroup relations

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: File:CDel node c1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png [6,3,3], File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png [3,6,3], File:CDel node.pngFile:CDel 6.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 6.pngFile:CDel node.png [6,3,6], File:CDel branch c1.pngFile:CDel split2.pngFile:CDel node c1.pngFile:CDel 6.pngFile:CDel node.png [6,3^[3]] and [3^[3,3]] File:CDel branch c1.pngFile:CDel splitcross.pngFile:CDel branch c1.png, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^*] (remove 3 mirrors, index 24 subgroup); [3,6,3^*] or [3^*,6,3] (remove 2 mirrors, index 6 subgroup); [1⁺,6,3,6,1⁺] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3^[3,3]]. The ringed Coxeter diagrams are File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png, File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png and File:CDel branch 11.pngFile:CDel splitcross.pngFile:CDel branch 11.png, representing different types (colors) of hexagonal tilings in the Wythoff construction.

Related polytopes and honeycombs

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.

It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.

It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:

Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,3} or t1{6,3,3}
Coxeter diagrams	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h0.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png ↔ File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{3,3} File:Uniform polyhedron-33-t2.png r{6,3} File:Uniform tiling 63-t1.png or File:Uniform tiling 333-t12.png
Faces	triangle {3} hexagon {6}
Vertex figure	File:Rectified order-3 hexagonal tiling honeycomb verf.png triangular prism
Coxeter groups	${\bar{V}}_3$, [3,3,6] ${\bar{P}}_3$, [3,3[3]]
Properties	Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t₁{6,3,3}, File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png half-symmetry construction alternates two types of tetrahedra. File:H3 633 boundary 0100.png

Hexagonal tiling honeycomb File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	Rectified hexagonal tiling honeycomb File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:Hyperbolic 3d hexagonal tiling.png	File:Hyperbolic 3d rectified hexagonal tiling.png
Related H2 tilings
Order-3 apeirogonal tiling File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	Triapeirogonal tiling File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png or File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.png
File:H2-I-3-dual.svg	File:H2 tiling 23i-2.pngFile:H2 tiling 33i-3.png

Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,3} or t0,1{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{3,3} File:Uniform polyhedron-33-t2.png t{6,3} File:Uniform tiling 63-t01.png
Faces	triangle {3} dodecagon {12}
Vertex figure	File:Truncated order-3 hexagonal tiling honeycomb verf.png triangular pyramid
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The truncated hexagonal tiling honeycomb, t_0,1{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure. File:H3 633-1100.png It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

File:H2 tiling 23i-3.png

Bitruncated hexagonal tiling honeycomb

Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png ↔ File:CDel node h0.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	t{3,3} File:Uniform polyhedron-33-t01.png t{3,6} File:Uniform tiling 63-t12.svg
Faces	triangle {3} hexagon {6}
Vertex figure	File:Bitruncated order-3 hexagonal tiling honeycomb verf.png digonal disphenoid
Coxeter groups	${\bar{V}}_3$, [3,3,6] ${\bar{P}}_3$, [3,3[3]]
Properties	Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,2{6,3,3}, File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure. File:H3 633-0110.png

Cantellated hexagonal tiling honeycomb

Cantellated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,3} or t0,2{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	r{3,3} File:Uniform polyhedron-33-t1.svg rr{6,3} File:Uniform tiling 63-t02.png {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Cantellated order-3 hexagonal tiling honeycomb verf.png wedge
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The cantellated hexagonal tiling honeycomb, t_0,2{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure. File:H3 633-1010.png

Cantitruncated hexagonal tiling honeycomb

Cantitruncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	t{3,3} File:Uniform polyhedron-33-t01.png tr{6,3} File:Uniform tiling 63-t012.svg {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Cantitruncated order-3 hexagonal tiling honeycomb verf.png mirrored sphenoid
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t_0,1,2{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure. File:H3 633-1110.png

Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	{3,3} File:Uniform polyhedron-33-t0.png {6,3} File:Uniform tiling 63-t0.svg {}×{6}File:Hexagonal prism.png {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Runcinated order-3 hexagonal tiling honeycomb verf.png irregular triangular antiprism
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The runcinated hexagonal tiling honeycomb, t_0,3{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure. File:H3 633-1001.png

Runcitruncated hexagonal tiling honeycomb

Runcitruncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,3{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	rr{3,3} File:Uniform polyhedron-33-t02.png {}x{3} File:Triangular prism.png {}x{12} File:Dodecagonal prism.png t{6,3} File:Uniform tiling 63-t01.png
Faces	triangle {3} square {4} dodecagon {12}
Vertex figure	File:Runcitruncated order-3 hexagonal tiling honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t_0,1,3{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure. File:H3 633-1101.png

Runcicantellated hexagonal tiling honeycomb

Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,2,3{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	t{3,3} File:Uniform polyhedron-33-t12.png {}x{6} File:Hexagonal prism.png rr{6,3} File:Uniform tiling 63-t02.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Runcitruncated order-6 tetrahedral honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,3{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure. File:H3 633-1011.png

Omnitruncated hexagonal tiling honeycomb

Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{6,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	tr{3,3} File:Uniform polyhedron-33-t012.png {}x{6} File:Hexagonal prism.png {}x{12} File:Dodecagonal prism.png tr{6,3} File:Uniform tiling 63-t012.svg
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Omnitruncated order-3 hexagonal tiling honeycomb verf.png irregular tetrahedron
Coxeter groups	${\bar{V}}_3$, [3,3,6]
Properties	Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,3{6,3,3}, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure. File:H3 633-1111.png

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs
• Alternated hexagonal tiling honeycomb

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [3]

External links

• John Baez, Visual Insight: {6,3,3} Honeycomb (2014/03/15)
• John Baez, Visual Insight: {6,3,3} Honeycomb in Upper Half Space (2013/09/15)
• John Baez, Visual Insight: Truncated {6,3,3} Honeycomb (2016/12/01)